PHYSICAL REVIEW A99, 063621 (2019)math.mit.edu/~dunkel/Papers/2019HeBuDu_PRA.pdfHEINONEN, BURNS, AND DUNKEL PHYSICAL REVIEW A 99, 063621 (2019) g 2j = 0 otherwise. The authors of Ref.

PHYSICAL REVIEW A 99, 063621 (2019)

Quantum hydrodynamics for supersolid crystals and quasicrystals

Vili Heinonen,1,* Keaton J. Burns,2 and Jörn Dunkel1,†

1Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307, USA2Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA

(Received 3 August 2018; published 24 June 2019)

Supersolids are theoretically predicted quantum states that break the continuous rotational and translationalsymmetries of liquids while preserving superfluid transport properties. Over the last decade, much progress hasbeen made in understanding and characterizing supersolid phases through numerical simulations for specificinteraction potentials. The formulation of an analytically tractable framework for generic interactions still posestheoretical challenges. By going beyond the usually considered quadratic truncations, we derive a systematichigher-order generalization of the Gross-Pitaevskii mean-field model in conceptual similarity with the Swift-Hohenberg theory of pattern formation. We demonstrate the tractability of this broadly applicable approach bydetermining the ground-state phase diagram and the dispersion relations for the supersolid lattice vibrationsin terms of the potential parameters. Our analytical predictions agree well with numerical results from directhydrodynamic simulations and earlier quantum Monte Carlo studies. The underlying framework is universal andcan be extended to anisotropic pair potentials with a complex Fourier-space structure.

DOI: 10.1103/PhysRevA.99.063621

I. INTRODUCTION

Supersolids are superfluids in which the local density spon-taneously arranges in a state of crystalline order. The existenceof supersolid quantum states was proposed in the late 1960sby Andreev, Lifshitz, and Chester [1,2] but the realization inthe laboratory has proven extremely difficult [3]. Recent ex-perimental breakthroughs in the control of ultracold quantumgases [4–7] suggest that it may indeed be possible to designquantum matter that combines dissipationless flow with theintrinsic stiffness of solids. Important theoretical insightsinto the expected properties of supersolids and experimentalcandidate systems have come from numerical mean-fieldcalculations and quantum Monte Carlo (qMC) simulationsfor specific particle interaction potentials [8–12]. What isstill lacking, however, is a general analytically tractableframework that allows the simultaneous characterization ofwhole classes of potentials as well as the direct computationof ground-state phase diagrams and dispersion relations forsupersolid lattice vibrations in terms of the relevant potentialparameters.

To help overcome such conceptual and practical limita-tions, we introduce here a generalization of the classicalGross-Pitaveskii (GP) mean-field model [13,14] by drawingguidance from the Swift-Hohenberg theory [15], which usesfourth-order partial differential equations (PDEs) to describepattern formation in Rayleigh-Bénard convection. Our ap-proach is motivated by the successful application of higher-than-second-order PDE models to describe classical solidifi-cation phenomena [16–18], electrostatic correlations in con-centrated electrolytes and ionic liquids [19,20], nonuniformFFLO superconductors [21], symmetry breaking in elastic

*[email protected]†[email protected]

surface crystals [22], and active fluids [23–25]. Whereashigher PDEs are often postulated as effective phenomenologi-cal descriptions of systems with crystalline or quasicrystallineorder [26], it turns out that such equations can be deriveddirectly within the established GP framework. The resultingmean-field theory yields analytical predictions that agree withdirect quantum hydrodynamic and recent qMC simulations[12,27] and specify the experimental conditions for realizingperiodic supersolids and coexistence phases, as well as super-solid states exhibiting quasicrystalline symmetry (Sec. V).

II. MEAN-FIELD THEORY

As in the classical GP theory [13,14], we assume that asystem of spinless particles can be described by a complexscalar field �(t, x) and that quantum fluctuations about themean density n(t, x) = |�|2 are negligible. Considering anisotropic pairwise interaction potential u(|x − x′|), the totalpotential energy density is given by a spatial convolutionintegral which can be expressed as a sum over Fourier-modecontributions ∝ u|n|2, where hats denote Fourier transforms(see Appendix A for a detailed derivation). If u is isotropicwith finite moments, then its Fourier expansion can be writtenas u(k) = ∑∞

j=0 g2 jk2 j [28], where k = |k| is the modulusof the wave vector, yielding the potential energy density12 n

∑∞j=0(−1) jg2 j (∇2) jn, with ∇2 denoting the spatial Lapla-

cian. The constant Fourier coefficients g2 j encode the spatialstructure of the potential. Variation of the total energy func-tional with respect to � yields the generalized GP equation(Appendix A)

ih∂t� =⎡⎣− h2

2m∇2 +

⎛⎝ ∞∑

j=0

(−1) jg2 j (∇2) j |�|2⎞⎠⎤⎦�. (1)

The classical GP model, corresponding to repulsive pointinteractions u = g0δ(x − x′), is recovered for g0 > 0 and

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HEINONEN, BURNS, AND DUNKEL PHYSICAL REVIEW A 99, 063621 (2019)

g2 j = 0 otherwise. The authors of Ref. [29] studied the caseg0, g2 > 0 and g2 j�4 = 0, keeping partial information aboutlong-range (k → 0) hydrodynamic interactions by effectivelyadding a surface energy term ∝ |∇n|2 to the energy density.However, as we see shortly, to describe supersolid states, thelong-wavelength expansion has to be carried out at least toorder k4.

To show this explicitly, it is convenient to express thecomplex dynamics, (1), in real Madelung form [31]. Writing� = √

n exp(iS) and defining the irrotational velocity fieldv = (h/m)∇S, Eq. (1) can be recast as the quantum hydro-dynamic equations (Appendix C; see also [32])

∂t n = −∇ · (nv), (2)

m(∂t + v · ∇ )v = −∇(δU/δn), (3)

with the effective potential energy U [n] to order O(k4),

U =∫

dx

[h2

8m

|∇n|2n

+ g0

2n2 + g2

2|∇n|2 + g4

2(∇2n)2

]. (4)

The first term is the kinetic quantum potential [31] and thekinetic energy is K = (m/2)

∫dx nv2. For nonleaky boundary

conditions, Eqs. (2) and (3) conserve the total particle numberN = ∫

dx n and energy E [n, v] = K[n, v] + U [n]. The quan-tum hydrodynamics defined by Eqs. (2)–(4) is Hamiltonianin terms of the momentum density mnv = hn∇S, therebydiffering, for example, from generalized Ginzburg-Landautheories for nonuniform FFLO superconductors [21] andfinite-temperature hydrodynamic approaches [1,33–35].

Local minima of E [n, v] have zero flow, v ≡ 0, and hencethe corresponding density fields must minimize U [n]. As-suming short-range repulsion g0 > 0, we see that uniformconstant-density solutions minimize U [n] when g2 � 0 andg4 = 0 (Appendix F); this case was studied in Ref. [29]. If,however, we consider the more general class of pair inter-action potentials with g2 ≷ 0, then short-wavelength stabil-ity at order k4 requires that g4 > 0. This situation arises,for example, for the two-dimensional (2D) Rydberg-dressedBose-Einstein condensate (BEC) studied in Ref. [27], whichhas g0 = 0.189h2/m, g2 = −0.113 (h2/m) μm2, and g4 =0.016 (h2/m) μm4 (Appendix G). Whereas for g2 > 0 rotonminima are absent as in the classical GP theory [8], supersolidground-state solutions [8,9] of Eq. (4) can exist when g2 < 0,as we see shortly.

To determine the ground-state phase diagram of U [n] withg2 < 0, it is convenient to define the characteristic wavenumber q0 = √−g2/(2g4) > 0, time scale t0 = m/(hq2

0 ), andenergy scale ε0 = h2q2

0/m. Focusing on the 2D case andadopting (q−1

0 , t0, ε0) as units henceforth, we can rewriteEq. (4) as

U [ρ] = 1

2

∫dx

[ |∇ρ|24ρ

+ αuρ2 + ρ(1 + ∇2)2ρ

], (5)

where ρ = mn(g4q20/h2) is the rescaled number density and

αu = 4g0g4

g22

− 1 (6)

the interaction parameter (Appendix C). In an infinite domain,the internal energy U [ρ] is completely parameterized by αu

and the average density ρ = ∫ρdx/

∫dx. For the Rydberg-

dressed BEC in Ref. [27], which has q0 = 1.87 μm−1, corre-sponding to a hexagonal lattice spacing of 3.88 μm, one findsαu = −0.043 and ρ ≈ 9.4 (Appendix G).

III. GROUND-STATE STRUCTURE

An advantageous aspect of the above framework is thatthe ground-state structure of U [ρ] in the (αu, ρ )-phase planecan be explored both analytically and numerically in a fairlystraightforward manner (Fig. 1). Standard linear stability anal-ysis (Appendix D 3) for uniform constant-density solutionspredicts a symmetry-breaking transition at αu = −1 whenρ � 1/8 and

αu = 1 − 16ρ

64ρ2when ρ > 1/8, (7)

indicated by the thin dashed line in Fig. 1(a). Refined analyt-ical estimates for the ground-state phases can be obtained byconsidering the Fourier ansatz ρ = ρ + ∑

j φ j exp (iq j · x).The pattern forming operator (1 + ∇2)2 penalizes modes with|q j | = 1, suggesting that single-wavelength expansions canyield useful approximations for the ground-state solutions.Conceptually similar studies for classical phase field models[18] imply that 2D and 1D close-packing structures realizinghexagonal and stripe patterns are promising candidates. Theone-mode approximation ansatz for a hexagonal lattice reads

ρ = ρ + φ0

3∑j=1

[exp (iq j · x) + c.c.], (8)

where the lattice vectors q j form the “first star” with qi ·q j = q2, if i = j and −q2/2 otherwise (c.c. denotes complexconjugate terms). Similarly, the stripe phase is defined byρ = ρ + [φ0 exp (iqx1) + c.c.]. These trial functions have tobe minimized with respect to the amplitudes φ0 and themagnitudes q of the reciprocal lattice vectors, which can bedone analytically (Appendix D 4). Our analytical calculationspredict four distinct pure ground-state types, which can beidentified as uniform (U), supersolid (SS), normal solid–like (NS), and droplet (D), and also a narrow U/SS coex-istence phase via Maxwell construction [dark-brown domainin Fig. 1(a)]. These naming conventions are directly adoptedfrom Ref. [12]. As always, mean-field predictions should besupplemented with other methods to properly characterize thesupersolid-NS transition to ensure that the wave functions arelocalized in the NS state [36]. Yet the comparison with theqMC simulations in Ref. [12] suggests that our mean-fieldresults capture essential aspects of their numerical results; seeinset in Fig. 1(a).

Using the one-mode approximation, the qualitatively dif-ferent states can be inferred as follows: The uniform U phasehas constant density ρ = ρ > 0, corresponding to φ0 = 0.The supersolid SS phase is distinguished through the exis-tence of energy-minimizing one-mode solutions with nonzeroamplitude φ0 that yield strictly positive periodic density pat-terns ρ > 0 everywhere. By contrast, in the NS state, theone-mode density field can become locally negative, signaling

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0.0

0.5

1.0

1.5

2.0

-1.2-1.0-0.8-0.6-0.4-0.20.0

SS

NS

UD

Ave

rage

den

sity

ρ

Interaction parameter αu

0.0

1.0

0.6

(a) (b)

0.2

0.4

0.8

(ρ−ρ

)/ρ

0

0.5

1

1.5

2

NS

U

Sca

led d

ensity

R

c2

Interaction strength U

SS

0 20 40 60 80 100

FIG. 1. (a) Phase diagram showing the ground-state structure of the potential energy functional U [ρ] in Eq. (5), calculated analyticallyusing a one-mode approximation. Inset: Phase diagram for an interacting Rydberg-dressed BEC obtained from qMC simulations, adaptedwith permission from Fig. 2 in Ref. [12]. (b) Examples of numerically computed minima of U [ρ] for parameters indicated by the yellowsymbols in (a). The uniform superfluid state (U) is stable against perturbations below the thin dashed red line. The hexagonal supersolid (SS;∗; Supplemental Material Movie 1 [30]) phase has a lower energy than the metastable supersolid stripe phase (SS; ‖). The first-order transitionbetween the U phase and the SS phase supports a narrow coexistence region (dark brown) with the uniform subphase having a lower density(x). In the NS state, the one-mode minimization of U yields locally negative density solutions, signaling failure of the approximation (see alsoFig. 2). In the droplet phase (•) no real-valued one-mode solutions exist. The single-droplet solution (D; •) is shown as an inset for scale in theother panels.

a breakdown of the one-mode approximation in this regime(Fig. 2). The dotted curve in Fig. 1(a) indicates where theone-mode approximation becomes invalid. The comparisonwith the qMC simulations in Ref. [12] shown in the insetin Fig. 1(a) suggests that this curve can be used as anapproximate predictor for the boundary of the SS phase.

2

4

6

8

10

12

14

0

Rel

ativ

enum

ber

den

sity

ρ/ρ

Relative position x/ann

FIG. 2. Difference between supersolid (SS) and normal solid–like (NS) states. One-dimensional cross sections of the density fieldsof two numerically determined ground states with the same averagedensity ρ = 0.4 but αu = −0.484 (SS; orange lines) and −0.60 (NS;blue lines), respectively. In both cases, the positions are rescaledwith the nearest-neighbor distance ann = 4π/

√3q to compensate the

different lattice spacings. Inset: Full 2D density field for αu = −0.6;the dashed red line indicates the 1D cross section.

An in-depth analysis of the NS phase, which is beyondthe scope of this paper, would require going beyond thetruncated mean-field model considered here. Similarly, inthe droplet D phase, no real-valued density solutions existwithin the one-mode approximation. To test the analyticalpredictions and explore the validity of the underlying one-mode approximation in detail, we performed a numericalground-state search (Appendix H) for various parameter pairs(ρ, αu). Representative examples of numerically found statesare shown in Figs. 1(b) and 2 and generally agree wellthe analytical predictions. The hexagonal supersolid at ρ =1.5, αu = −0.2 [asterisk in Fig. 1(b)] is indeed dominatedby a single mode, with the second-highest mode being ∼20times smaller. Our analytical one-mode estimates suggestthat the supersolid stripe states have a higher energy thanthe hexagonal states, with the energy difference going to 0as one approaches the U-SS phase boundary [solid line inFig. 1(a)]. This opens the possibility that systems at nonzerokinetic energy or in a strained geometry might arrange ina stripe configuration similar to the metastable state shownat ρ = 0.5, αu = −0.45 [parallel lines in Fig. 1(b)]. Thenumerical solution for αu = −0.2178, ρ = 1.0 [x in Fig. 1(b)]confirms the analytically predicted coexistence of uniformand supersolid phases, suggesting that the one-mode approx-imation places the coexistence region accurately in the phasediagram. For coexistence to be observable in experiments, theenergy of the uniform and supersolid bulk regions has to besignificantly higher than that of the interface, which requiresa sufficiently large system size. At high ρ, the coexistence gapcloses, approaching the asymptote αu = −0.22/ρ of the phasetransition.

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Our simulations show that the one-mode approximationdescribes the ground-state structure accurately down to av-erage densities ρ � 0.4 near the uniform-supersolid phasetransition. If ρ and/or αu are reduced further, higher modeswill no longer be negligible. The difference between a super-solid one-mode ground state and a normal solid multimodesolution at the same average density ρ = 0.4 is shown inFig. 2. The density profile for the NS state resembles acollection of Gaussian peaks, which typically have a signifi-cantly larger separation than the peaks of a one-mode solutionat similar ρ. Moreover, the very low-density values in theregions between the maxima suggest that small quantum orthermal fluctuations may suffice to destroy phase coherencein this regime. Finally, the domain αu < −1 corresponds toa negative GP parameter g0 < 0, thus representing attractivecontact interactions. The numerically determined ground stateat ρ = 0.1, αu = −1.1 [filled circle in Fig. 1(b)] realizes asingle droplet with an approximately Gaussian density profile,qualitatively similar to recent experimental observations ofquantum droplet formation in dilute 39K BECs [37].

IV. DYNAMICS

We next describe how the above framework can be usedto obtain predictions for the sound-wave dynamics in a su-persolid (see Appendix E for details of calculations). Thesupersolid phase breaks the continuous translational sym-metry and supports lattice vibrations. These vibrations canbe studied analytically close to the uniform-supersolid phasetransition, where the one-mode approximation for the densityρ is accurate. Near the phase transition the local deviationof ρ from its mean value ρ is relatively small, and one canTaylor-expand the nonlinear quantum potential in ρ aroundρ. Adopting this approximation, we now consider a changeof coordinates x → x − u(t, x), where u is a displacementfield in the Eulerian frame x. Since we are interested inhydrodynamic long-wavelength sound excitations, we mayassume that the displacement field varies over a length scalesignificantly larger than the spacing between the hexagonaldensity peaks |∇u| � 1. Inserting the one-mode ansatz (8)into Eq. (5), and keeping only the leading terms in u, oneobtains an energy functional U [u] that is quadratic in thedisplacement field (Appendix E). The approximative dynam-ics of u is found from Eqs. (2) and (3) by noting [38] that,for small displacements, ∂t u = v and ∂t v = −∇δU/δρ =−(1/ρ )δU/δu hold. This gives the linear equation

∂2t u = 3φ2

0q2

ρ

[1 − φ0/ρ + 5φ2

0/ρ2

4ρ∇2u

+ (3q2 − 2)∇2u + 2q2∇(∇ · u) − 2

3∇4u

], (9)

which is solved by a plane wave u0 exp [i(k · x − ωt )]. Sincethe field v ∝ ∇S describes an irrotational potential flowwith ∇ × v = 0, only longitudinal modes are allowed.Inserting the plane-wave ansatz in Eq. (9) yields thenonlinear dispersion relation ω/ω‖ = (k/k‖)

√1 + (k/k‖)2,

with k2‖ = 3

2 {[(5φ20/ρ

2 − φ0/ρ + 1)/4ρ] − 2 + 5q2} andω2

‖ = 2q2φ20k4

‖/ρ, which is shown as the blue line in Fig. 3(a).For supersolids described by the one-mode approximation,

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.00.0

0.06

0.06

-4 -2 0 2 4

(a)

(b)

0.5 1.0 1.5 2.0 2.5

ρ10

5 x|v|

Fre

quen

cyω/ω

Wave vector k/k

FIG. 3. (a) The analytically calculated nonlinear dispersion re-lation (blue line) for plane waves becomes asymptotically linear forsmall k values (dashed orange line). See text for the scaling factors ω‖and k‖. Inset: The analytical prediction (blue line) agrees well withnumerical results (symbols) obtained by solving the full nonlineardynamical system described by Eqs. (2) and (3) for ρ = 1.0 and αu =−0.24. (b) Representative snapshot from a numerical simulationshowing the magnitude of the velocity field and the density field.See also Movie 2 (Supplemental Material [30]).

one has q ≈ 1 and φ0/ρ < 1/3, implying k/k|| � 1. Thissuggests that vibrations with wavelengths larger than thelattice constant generally exhibit a linear dispersion and thatthe speed of sound is simply given by c‖ = ω‖/k‖ [dashedorange curve in Fig. 3(a)].

The analytical prediction derived from the linear Eq. (9)agrees well with the numerical dispersion results obtained bysimulating the full nonlinear Eqs. (2) and (3) with the open-source spectral code DEDALUS [39]; see inset in Fig. 3(a).In our simulations, we analyzed a longitudinal mode in aperiodic box, for four box sizes in the direction of thewave, using ρ = 1.0 and αu = −0.24 (Appendix H). Forthese parameters, one finds φ0 = 0.266 and q = 0.934 byminimizing the energy U of the supersolid state with the one-mode approximation. Representative simulation snapshotsshowing the magnitude of the velocity field and the densityare shown in Fig. 3(b); see also Movie 2 (SupplementalMaterial [30]). The low-k mode is visible as an envelopingmodulation of the velocity field. The sound speed was mea-sured in the simulations by estimating w‖ and k‖ from alinear fit, yielding c‖ = 0.6994 ± 0.0015, in excellent agree-ment with the theoretically predicted value ω‖/k‖ = 0.6995.Thus, although waves can scatter from inhomogeneities ofthe density field in the fully nonlinear system, the long-wavelength dynamics is accurately captured by the linearizedtheory.

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0.0

1.0

( ρ−ρ

) /ρ

0.5

FIG. 4. Superfluid ground state exhibiting quasicrystalline order,as predicted by Eq. (1) for an interaction potential u having relevantterms up to eighth order in the expansion (10). System param-eters: ρ = 1.0, αu = −0.9, b1 = 0.09, b2 = 0, a1 = 0.982, a2 =2 cos (π/12). Here ρmin ≈ 0.395 and ρmax ≈ 2.969. The minimumdensity is well above 0, suggesting that this state is superfluid. Inset:Absolute value of the Fourier-transformed density, |ρ|, indicating a12-fold rotational symmetry with two rings in the reciprocal lattice.See also Movie 3 (Supplemental Material [30]).

V. SUPERFLUID QUASICRYSTALS

The supersolid phase in Fig. 1 is caused by the pat-tern forming term αu(1 + ∇2)2 in the internal energy U .This fourth-order contribution makes plane waves with wavenumber k0 = |k| = 1 energetically favorable, giving rise to ahexagonal pattern. More complex supersolid structures canbe expected in systems where the Fourier-transformed inter-action potential u possesses multiple local minima ki, whereu(ki ) < 0. To demonstrate that this is indeed the case, let usreplace (1 + ∇2)2 with[

b1 + (a2

1 + ∇2)2][b2 + (

a22 + ∇2)2]

, (10)

which corresponds to keeping terms up to order j = 4 inthe generalized GP equation, (1). The resulting eighth-orderoperator in Eq. (10) is energetically bounded from belowand accounts for an asymmetry in the local potential energyminima ki through the parameters bi. Note that ai and ki

are in general no longer equal when bi = 0. With a largernumber of physically relevant length scales at play, systemscan attain a wider range of ground-state structures [40,41].To illustrate this, we computed the ground state of Eq. (1)for the expansion (10) with parameters ρ = 1.0, αu = −0.9,b1 = 0.09, b2 = 0, a1 = 0.982, a2 = 2 cos (π/12). As evidentfrom the corresponding mean-field density ρ and its absoluteFourier transform |ρ| in Fig. 4, the ground state in this caseis a quasicrystalline superfluid state with 12-fold rotationalsymmetry; see also Movie 3 (Supplemental Material [30]).

In this context, we mention that earlier discussions [42,43]of superfluid quasicrystalline states considered spin and pseu-dospin interactions with only one relevant length scale. Bycontrast, the quasicrystalline ground state in Fig. 4 formsdue to the competition between the two length scales set bythe local minima of the interatomic potential u. While thismechanism is reminiscent of classical quasicrystal pattern

formation [40,41], the mean-field quantum systems discussedhere differ from their classical counterparts through the pres-ence of the quantum potential term which penalizes high-kmodes.

VI. CONCLUSION

To conclude, we have introduced and studied a higher-order generalization of the classical GP mean-field theorythat accounts for the structure of pair interactions throughthe relevant Fourier coefficients of the underlying potential.The resulting hydrodynamic equations share many concep-tual similarities with classical Swift-Hohenberg-type patternformation models [15], for which a wide range of advancedmathematical analysis tools exists [18,44,45]. The generalizedGP equation, (1), allows us to transfer these techniques di-rectly to quantum systems. By focusing on the fourth-ordercase, we obtained analytic predictions for ground-state phasediagrams and low-energy excitations that agree well withdirect numerical simulations. With regard to experiments, ourresults suggest that the coexistence phase at the first-ordersuperfluid-supersolid transition may be the most promisingregime for observing supersolids. We expect this mean-fieldprediction to be robust, as the analytically derived uniform-supersolid phase transition curve (solid line in Fig. 1) agreeswell with recent qMC simulations [12] that account for be-yond mean-field effects [Fig. 1(a), inset].

When the interaction parameter αu in Eq. (6) becomestoo small, a transition to a normal solid state is expected,as quantum fluctuations will likely destroy phase coherencein the low-density domains. However, grain boundaries ofpolycrystalline normal solid could still show superfluid be-havior [46,47] and can be studied using the theory presentedhere. Interestingly, our numerical ground-state analysis sug-gests that at low average densities a quantum droplet can bestabilized by the kinetic quantum potential without requiringhigher-order nonlinearities to correct for quantum fluctuations[37,48]. From a conceptual perspective, the generalized GPframework appears well suited for future extensions: By cal-culating the quasicrystalline ground state (Fig. 4 and Supple-mental Material Movie 3 [30]), we have already demonstratedhow to extend the approach to systems with multiple relevantlength scales. The theory can be refined by including Lee-Huang-Yang [49,50] corrections ∝ ρ5/2 that penalize sharplypeaked densities. Generalizations to anisotropic potentialsand multicomponent systems seem both feasible and exper-imentally relevant [4,37,51]. We thus hope that the abovediscussion can provide useful guidance for future efforts inthese and other directions.

ACKNOWLEDGMENTS

This work was supported by the Finnish Cultural Founda-tion (V.H.) and an Edmund F. Kelly Research Award (J.D.).

APPENDIX A: DERIVATION OF THEMEAN-FIELD ENERGY

1. Mean-field reduction of the many-particleSchrödinger equation

For completeness, we present the derivation of the gen-eralized GP equation, (1), from a many-particle Schrödinger

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HEINONEN, BURNS, AND DUNKEL PHYSICAL REVIEW A 99, 063621 (2019)

equation with a pair potential u(ri, r j ). Our starting point isa bosonic N-particle quantum system, described by a wavefunction �N (t, r1, . . . , rN ) that satisfies the exchange symme-try

�N (t, . . . , ri, . . . , r j, . . .)

= +�(t, . . . , r j, . . . , ri, . . .). (A1)

The dynamics of the system is governed by the N-particleSchrödinger equation

ih∂t�N = HN�N , (A2)

where the N-particle Hamiltonian is defined by

HN = −α∑

i

∇2ri

+∑

i

vext(ri ) + 1

2

N∑i=1

N∑j =i

u(ri, r j ),

(A3)

with α = h2/(2m). Equation (A2) can be written as a variationof the expectation value of HN ,

ih∂t�N = δE

δ�∗N

, (A4)

where

E�N [�N , �∗N ] := 〈HN 〉 =

∫drN�∗

N HN�N . (A5)

The key step in the derivation of the generalized GP equationis the mean-field approximation

�N (t, r1, . . . , rN )

= 1

N!

∑π∈SN

ψ (t, rπ (1) )ψ (t, rπ (2) ) . . . ψ (t, rπ (N ) ), (A6)

where the sum is over the permutation group SN of indices1, . . . , N . This approximation is equivalent to assuming thatthere are no entangled particles in the system. It also impliesthat the one-particle probability distributions ψ∗(ri )ψ (ri ) areindependent and identical.

We can use Eq. (A6) to calculate the mean-field approxi-mation of the energy. The symmetric form of the mean-fieldansatz allows for relabeling of indices in the evaluation of theHamiltonian HN , giving

E� =∫

dr(−α�∗∇2� + |�|2vext ) + N (N − 1)

2N2

×∫

dr1dr2[�∗(r1)�(r1)u(r1, r2)�∗(r2)�(r2)].

(A7)

Here � = √Nψ so |�|2 becomes the number density n. For

large N , one has N (N − 1)/N2 ≈ 1.To derive a dynamical equation for the one-particle wave

function, we evaluate the expectation value of the time evo-lution operator. Multiplying Eq. (A2) by �∗

N and integrating

over the spatial coordinates gives

〈ih∂t 〉 =∫

drN�∗N ih∂t�N

=∫

drN�∗N ih

N∑j=1

∂�N

∂�(r j )∂t�(r j )

= 1√N

∫drN�∗

N ihN∑

j=1

�N

�(r j )∂t�(r j )

= 1

N

N∑j=1

∫dr j�

∗(r j )ih∂t�(r j )

=∫

dr�∗(r)ih∂t�(r) = E�.

(A8)

Taking the functional derivative with respect to �∗ yields

ih∂t� = δE�

δ�∗ , (A9)

where the right-hand side is given by

δE�

δ�∗ =(

−α∇2 + vext +∫

dr′|�(r′)|2u(r, r′))

�. (A10)

The functional Gross-Pitaevskii equation in Eq. (A9) wasdiscussed by Gross in his seminal paper on superfluid vortices[13]. This theory corresponds to a quantum density functionaltheory where the exchange-correlation energy is neglected;for strongly correlated systems, see, e.g., Ref. [52].

2. Fourier expansion for isotropic potentials

The energy contribution for an isotropic pair potential u is

〈u〉 = 1

2

∫∫dr1dr2[�∗(t, r2)�∗(t, r1)

× u(|r2 − r1|)�(t, r1)�(t, r2)], (A11)

which can be rewritten as

〈u〉 = 1

2

∫dr[n(t, r)(u ∗ n)(t, r)], (A12)

where (u ∗ n) is the convolution∫dr′[u(|r − r′|)n(t, r′)] (A13)

and n is the local number density |�|2.The Fourier transform of a pair potential

u(k) =∫

dr[e−ik·ru(r)] (A14)

with finite moments can be expressed as a power series in k2

as

u(k) =∞∑j=0

g2 jk2 j . (A15)

Using this the convolution of the pair potential becomes

(u ∗ n) = F−1[un] =∞∑j=0

(−1) jg2 j∇2 jn. (A16)

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QUANTUM HYDRODYNAMICS FOR SUPERSOLID CRYSTALS … PHYSICAL REVIEW A 99, 063621 (2019)

Here F−1 denotes the inverse Fourier transform.Now

〈u〉 = 1

2

∫dr

⎡⎣n

⎛⎝ ∞∑

j=0

(−1) jg2 j∇2 j

⎞⎠n

⎤⎦. (A17)

The energy E� becomes

E� [�∗, �]

= 〈u〉 +∫

dr[

h2

2m|∇�|2 + vext|�|2

](A18)

and the dynamics is given by Eq. (A9).The interaction term [Eq. (A17)] can be effectively trun-

cated at low j because the kinetic energy contribution ∝|∇�|2 penalizes high-order Fourier modes, forcing their am-plitudes to be small. Let us examine the lowest-order trunca-tions of Eq. (A17):

(i) Zeroth order: This case corresponds to the standard GPequation.

(ii) Second order: This case was studied in Ref. [29]. Thecoefficient g2 must be nonnegative to ensure that the energyis bounded from below, thus leading to a penalization ofvariations in n. As shown in Appendix F, for repulsive contactinteractions (g0 > 0) the only possible ground state is theuniform state characterized by a constant density n(t, r) = n.

(iii) Fourth order: In Appendix D we show that this orderleads to pattern formation.

(iv) Higher orders: Interaction potentials with multiplecompeting length scales allow the realization of more complexsymmetries, including honeycomb pattern or quasicrystals[40,41]; see example in Sec. V.

3. Fourth-order expansion

Let us reparametrize the contribution due to the interactionpotential as

〈u〉 =∫

dr[ur

2n2 + ue

2n(∇2 + q2

0

)2n]. (A19)

In terms of the wave function � we have

ih∂t� = H�, (A20)

where

H = − h2

2m∇2 + vext + [

ur + ue(∇2 + q2

0

)2](A21)

is the system Hamiltonian.The parameters are defined as

q20 = − g2

2g4, ue = g4, ur = g0 − g2

2

4g4. (A22)

For negative values of q20, no pattern forming is to be expected.

The sign of q20 is fully determined by g2 since g4 > 0 in order

to ensure finite energy for small-wavelength Fourier modes.

APPENDIX B: UNITS

In an effort to make the text more readable we use some ofthe same symbols for both SI and reduced units, whereas inthe Appendixes we have tried to maintain mathematical rigor,

TABLE I. Units and notation. Here d is the dimension of thesystem.

Symbol Description Units Variable(s) Text

r Length m – xt Time s – t� Wave function m−d/2 (t, r) �

n Number density m−d (t, r) nu Interaction potential J md r = |r| ux Length q−1

0 – xτ Time m/(hq2

0 ) – ts Minimization parameter m/(hq2

0 ) – – Wave function h/(q0

√mue ) (τ, x) –

ρ Number density h2/(mueq20 ) (τ, x) ρ

U Effective potential energy h2q20/m ρ U

u Displacement field q−10 (τ, x) u

with the possible drawback of having too many symbols inthe text. In the text v denotes the velocity field both in SI unitsand in units of hq0/m, whereas in the Appendixes the field v isonly in units of hq0/m. In the Appendixes we use the notationF� for integral energies, where � is the field variable. We haveomitted the field variable for functionals of the fields ρ and v.In the text the same uppercase characters of the form F areused for energy integrals in both SI units and reduced units. Inboth the text and the Appendixes the integral energies that arefunctionals of reduced fields are also in energy units that arereduced. See Table I for a partial list of symbols used in thetext and the Appendixes.

APPENDIX C: HYDRODYNAMIC FORMULATION

The wave function can be rewritten using the polardecomposition as

(τ, x) = R(τ, x) exp (iS(τ, x)). (C1)

Now ρ = | |2 = R2. We define v := ∇S, allowing us to writeEq. (A20) as a set of flow equations,

DvDt

= −∇[Q + vext + (αu + (∇2 + 1)2)ρ], (C2a)

∂τρ = −∇ · [ρv], (C2b)

where D/Dτ = ∂τ + v · ∇ is the advective time derivative,αu = ur/(ueq4

0 ), and

Q = −1

2

∇2√ρ√ρ

(C3)

is the quantum potential. Equation (C2a) can be expressed interms of a functional derivative as

DvDt

= −∇ δU

δρ, (C4)

where

U [ρ] =∫

dx[ |∇ρ|2

8ρ+ vextρ + αu

2ρ2 + 1

2ρ(∇2 + 1)2ρ

](C5)

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HEINONEN, BURNS, AND DUNKEL PHYSICAL REVIEW A 99, 063621 (2019)

is the effective potential energy. These dynamics conserve thetotal energy of the system

E [ρ, v] = K[ρ, v] + U [ρ], (C6)

where

K[ρ, v] =∫

dx[

1

2ρ(τ, x)|v(τ, x)|2

](C7)

is the kinetic energy.

APPENDIX D: GROUND STATES

In this section we discuss ground states of the systemdescribed by energy E [Eq. (C6)]. We define dissipativeprocesses that are used to minimize E and use them to analyzethe ground-state structure of the system both analytically andnumerically.

1. Dissipative dynamics in the density formalism

The energy E is locally minimized when v = 0 and U isminimized with respect to ρ. Here we present two types ofdensity preserving dissipative flows parametrized by s thatminimize U .

The first one is given by

∂sρ(s, x) = − δU [ρ]

δρ(s, x)+ λ(s)

= −[Q + (αu + (∇2 + 1)2)ρ] + λ(s), (D1)

where

λ = −∫

dx[

δU [ρ]

δρ

](D2)

is a Lagrange multiplier that keeps the total density∫

ρdxconstant in time. Here −

∫denotes the integral mean defined by

−∫

f dx = ∫f dx/

∫dx. The process described by Eq. (D1) is

referred to as nonlocal dissipative dynamics.Local dissipative dynamics is defined by

∂sρ = ∇2 δU [ρ]

δρ

= ∇2[Q + (αu + (∇2 + 1)2)ρ]. (D3)

It can be shown that both these dynamics will lead to anonincreasing U , i.e., ∂sU (s) � 0.

2. Dissipative dynamics in the wave formalism

Let us define the total energy in the wave formalism:

E [ ∗, ]

= 1

2

∫dx{|∇ |2 + αu| |4 + [(1 + ∇2)| |2]2}. (D4)

The generalized nonlinear Schrödinger’s equation can be writ-ten as

∂s = −iδE

δ ∗

= −i

[−1

2∇2 + (αu + (∇2 + 1)2)| |2

] . (D5)

The time evolution can be made dissipative by making amodification

∂s (s, x) = −(i + μ)δE

δ ∗ + λ(s) (s, x), (D6)

where μ is some positive dissipation rate. Here λ(t ) is aLagrange multiplier ensuring the global conservation of thedensity | |2 and can be calculated as

λ(s) = μ

N

∫dx

[ ∗(s, x)

δE

δ ∗

]= μ

N〈H〉, (D7)

where

N =∫

dx[| (s, x)|2] (D8)

is the rescaled number of particles in the system and H is theHamiltonian operator. This can be interpreted as the expectedenergy per particle times the dissipation rate.

The dynamics can be made overdamped by dropping theimaginary part of the mobility, giving

∂s (s, x) = −μδE

δ ∗ + λ(s) (s, x), (D9)

where λ is defined by Eq. (D7). This equation has no couplingbetween the real and the imaginary parts of allowing forsetting Im[ (s, x)] = 0, which gives (s, x) = √

ρ(s, x).Both Eq. (D6) and Eq. (D9) can be shown to lead to a

nonincreasing energy E with respect to parameter s.

3. Linear stability analysis

Let us analyze the stability of Eq. (D3) against small peri-odic perturbations about a constant state, i.e., ρ(τ, x) = ρ +ε(τ ) exp (iq · x), where ε is some positive, small, spatiallyindependent amplitude. Inserting this ansatz into Eq. (D3) andabbreviating q = |q| we find

∂τ ε(τ ) = −(

q4

4ρ+ αuq2 + q2(1 − q2)2

)ε(t )

= C(q)ε(τ ) (D10)

up to linear order in ε. Maximizing the coefficient C withrespect to q defines the most unstable perturbation. If sucha maximum exists, it suffices to examine the value of C at thegiven maximum. If no maximum exists, C is maximized byq = 0, which will give a stable constant solution.

Solving the aforementioned problem gives the conditionfor stability. The uniform solution is not stable against pertur-bations if

αu <

{−1, ρ � 1/8,

(1 − 16ρ )/(64ρ2

), ρ > 1/8.

(D11)

This defines the spinodal curve for the uniform phase. Thestability phase diagram with the value for the most unstable qis shown in Fig. 5.

4. One-mode approximation

The pattern forming part of the internal energy Eq. (C5)can be written in Fourier space as

1

8π2

∫dk[(1 − k2)2ρ(k)2] (D12)

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QUANTUM HYDRODYNAMICS FOR SUPERSOLID CRYSTALS … PHYSICAL REVIEW A 99, 063621 (2019)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

stable

unstable

Max

imal

ly u

nst

able

wav

e num

ber

q

ρA

vera

ge d

ensity

0.5 - 0.5 - 1.0 - 1.50.00.0

0.5

1.0

1.5

2.0

Interaction parameter αu

FIG. 5. Stability diagram of the uniform density state as a func-tion of the interaction parameter αu and average density ρ. Contourlines show the value for the most unstable wave number q of thelinear perturbation.

by using Plancherel’s theorem. For crystalline ground stateswe have

ρ(x) =∑k∈G

akeik·x, (D13)

where G is the point group of the reciprocal crystal lattice.The pattern forming term will penalize any modes with |k| =1, suggesting that including only the vectors k in the firstBrillouin zone might approximate well the ground state. Herewe analyze one-mode approximations of periodic numberdensities in two dimensions.

a. Stripe phase

We use the ansatz

ρS(x) = ρ + φ0nS(x) = ρ + φ0(eiq·x + c.c.)

= ρ (1 + A cos [q · x]), (D14)

where c.c. stands for the complex conjugate and A = 2φ0/ρ.We can simplify the calculations by choosing q = (q, 0) andwriting q · x = qx, leading to

ρS(x) = ρ + φ0nS(x) = ρ + φ0(eiqx + c.c.)

= ρ (1 + A cos[qx]).

Inserting the stripe ansatz ρS in Eq. (C5) gives an averageenergy

U = −∫

Cp

dx

[1

8

|∇ρS|2ρS

+ αu

2ρ2

S + 1

2ρS(∇2 + 1)2ρS

], (D15)

where Cp is the interval [0, 2π/q] corresponding to a period ofρS. The integrand is constant in the y coordinate, reducing theenergy to a 1D integral. We start by calculating the quadraticterm

ρ2S = ρ2 + 2ρφ0nS(x) + φ2

0nS(x)2.

All the terms that have an oscillating component exp(iq jx)with j = 1, 2, 3, . . . make no contribution to the averageenergy. Therefore the interesting terms are np

S with p > 1.

The terms without an oscillating component are referred to asresonant terms since they resonate with the linear operation∫

Cpdx.Now

−∫

Cp

dx(n2

S

) = −∫

Cp

dx(e2iqx + e−2iqx + 2) = 2

and

−∫

Cp

dx[ρS(x)2] = ρ2 + 2φ20 .

We also need

−∫

Cp

dx[ρS(x)∇2pρS(x)] = −∫

Cp

dx[(−1)pρS(x)q2pn(x)]

= 2(−1)pq2pφ20 .

The pattern forming part of the average energy becomes

−∫

Cp

dx

[αu

2ρS(x)2 + 1

2ρS(x)(∇2 + 1)2ρS(x)

]

= αu + 1

2ρ2 + ((1 − q2)2 + αu)φ2

0 . (D16)

The remaining part of the average energy is the part from thequantum potential

−∫

Cp

dx

[ |∇ρS(x)|2ρS(x)

]= −∫

Cp

dx

[(∂xnS(x))2

ρ + nS(x)

].

Here we use nS(x) = ρA cos ξ , where ξ = qx:

−∫

Cp

dx

[(∂xnS(x))2

ρ + nS(x)

]= A2q2ρ

2π

∫ 2π

0dξ

[sin 2ξ

1 + A cos ξ

]

= A2q2ρ

2π

∫ π

−π

dξ

[sin 2ξ

1 + A cos ξ

].

This can be solved with a substitution, p = tan (ξ/2). Thisgives

sin ξ = 2p

1 + p2, cos ξ = 1 − p2

1 + p2, dξ = 2

1 + p2d p.

Now

A2q2ρ

2π

∫ π

−π

dξ

[sin 2ξ

1 + A cos ξ

]

= A2q2ρ

2π

∫ ∞

−∞d p

⎡⎣

( 2p1+p2

)2

1 + A( 1−p2

1+p2

) 2

1 + p2

⎤⎦

= 8A2q2ρ

2π (1 − A)

∫ ∞

−∞d p

[p2(

p2 + γ 2A

)(1 + p2)2

],

where γ 2A = (1 + A)/(1 − A). Note that ρS � 0 requires A �

1. We also assume that A � 0, implying that γ 2A > 0 and that

063621-9

HEINONEN, BURNS, AND DUNKEL PHYSICAL REVIEW A 99, 063621 (2019)

Im x

Re x

x

FIG. 6. Semicircular contour in the complex plane used for cal-culating an integral on the real axis. The symbol x marks a singularityon the imaginary axis.

γA is real. The integral∫ ∞

−∞d p

[p2

p2 + γ 2A

1(1 + p2

)2

]

=∫ ∞

−∞d p

[p2

(p + iγA)(p − iγA)

1

(p + i)2(p − i)2

]

=:∫ ∞

−∞d p f (p)

can be solved in a standard way using residue theorem for asemicircular path on the complex plane shown in Fig. 6 andtaking the radius to ∞. For this we need residues at singu-larities in the upper-half complex plane (Im p > 0). There aretwo of these, p = iγA and p = i. The latter is of order 2:

Res ( f , iγA) = iγA

2(γ 2

A − 1)2 ,

Res ( f , i) = − i(1 + γ 2A )

4(γ 2

A − 1)2 .

Now∫ ∞

−∞d p f (p) = 2π i[Res ( f , iγA) + Res ( f , i)],

giving

8A2q2ρ

2π (1 − A)

∫ ∞

−∞d p

[p2

p2 + γ 2A

1

(1 + p2)2

]

= q2ρ(1 −√

1 − A2).

Finally,

U = q2ρ

8

⎛⎝1 −

√1 − 4

(φ0

ρ

)2⎞⎠

+ αu + 1

2ρ2 + ((1 − q2)2 + αu)φ2

0 . (D17)

b. Hexagonal phase

The crystalline hexagonal ground-state solution is approxi-mated similarly to the stripe phase with the one-mode approx-imation

ρH(r) = ρ + φ0nH(r)

= ρ + φ0

3∑j=1

(eiq j ·r + c.c.), (D18)

where the reciprocal lattice vectors q j have a hexagonalsymmetry, i.e.,

q j · qi ={

q2, i = j,− 1

2 q2, i = j.

Equation (D18) can be rewritten as

ρH = ρ

(1 + 2A cos

(√3qx

2

)cos

(qy

2

)+ A cos (qy)

),

where, again, A = 2φ0/ρ.Calculating the resonant terms for the polynomial part of

the internal energy is pretty straightforward [53]. For thequantum potential we have

−∫

Cp

dx[

1

8

|∇ρH|2ρH

d

]= −∫

Cp

dx

[ρq2A2

8

3 sin2(√

3qx2

)cos2

( qy2

) + (cos

(√3qx2

)sin

( qy2

) + sin(qy))2

2A cos(√

3qx2

)cos

( qy2

) + A cos(qy) + 1

],

where Cp is the primitive lattice cell shown in Fig. 7. Let√

3qx/2 = x and qy = y. Now this integral is

ρq2A2

32π2

∫ 2π

0

∫ 2π

0dxdy

[3 sin2 (x) cos2

( y2

) + (cos (x) sin

( y2

) + sin(y))2

2A cos (x) cos( y

2

) + A cos(y) + 1

].

The integration in x can be done by repeating the calculation done for the stripe phase. The integrand will have two purelyimaginary singularities that can be calculated by factoring the polynomial appearing in the denominator. This gives an integralin y over a period of 2π :

ρq2

32π

∫ π

−π

dy

[1

cos(y) + 1

(−2 − 6A + 10A2 + (9A2 − 4A − 4) cos(y) − 2A cos(2y) − A2 cos(3y)√

2A2 cos(2y) − 6A2 − 8(A − 1)A cos(y) + 4+ (A + 2) cos(y) + 2A + 1

)].

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QUANTUM HYDRODYNAMICS FOR SUPERSOLID CRYSTALS … PHYSICAL REVIEW A 99, 063621 (2019)

Cp

ann

FIG. 7. One-mode density field with a hexagonal symmetryshowing the primitive lattice cell Cp and the nearest-neighbor dis-tance ann = 4π/(q

√3).

Noting that cos(−y) = cos(y) and making the change ofvariables cos y = 1 − 2ξ gives

q2ρ

32π

∫ 1

0dξ

[P1(ξ )

ξ√

f1+ Q1(ξ )

ξ√

ξ (1 − ξ )

],

where

P1(ξ ) = −16A2ξ 3 + (24A2 − 8A)ξ 2

+ (4A − 4)ξ + (A − 1)2,

Q1(ξ ) = (2A + 4)ξ + A − 1,

and

f1(ξ ) = (4A2ξ 2 + (4A − 8A2)ξ + A2 − 2A + 1)(1 − ξ )ξ .

Let us first calculate the part

I1 :=∫ 1

0dξ

[Q1(ξ )

ξ√

ξ (1 − ξ )

]

=∫ 1

0dξ

[4 + 4A√ξ (1 − ξ )

]−∫ 1

0dξ

[1 − A

ξ√

ξ (1 − ξ )

].

Both parts can be calculated with trigonometric substitution.This gives

I1 = 2(2 + A)π − limξ→1

2(1 − A)ξ√ξ (1 − ξ )

= 2(2 + A)π − limε→0

2(1 − A)√

ε−1 − 1. (D19)

The second part diverges and has to be combined with otherterms. Now the other part needed here is

I2 :=∫ 1

0dξ

[P1(ξ )

ξ√

f1

]=∫ 1

0dξ

[−16A2ξ 2

√f1

− 8A(1 − 3A)ξ√f1

− 4(1 − A)√f1

+ (1 − A)2

ξ√

f1

]. (D20)

These are elliptic integrals. The only diverging part is∫ 1

0dξ

[(1 − A)2

ξ√

f1

].

This can be dealt with by using an identity,

(2 − s)a0Js−3 + 12 a1(3 − 2s)Js−2

+ a2(1 − s)Js−1 + 12 a3(1 − 2s)Js − sa4Js+1

=√

f (ξ − c)−s|10, (D21)

where

Js[ f ] =∫ 1

0dξ

[1√

f (ξ − c)s

], (D22)

f is a third- or fourth-order polynomial with

f = a0(x − c)4 + a1(x − c)3

+ a2(x − c)2 + a3(x − c) + a4,

s = 1, 2, 3, . . . , and c is some arbitrary constant [54]. In ourcase s = 1, c = 0, and f = f1, giving

a0 = −4A2, a1 = −4(1 − 3A)A, a2 = −(1 − 3A)2,

a3 = (1 − A)2, a4 = 0.

We have∫ 1

0dξ

[(1 − A)2

ξ√

f1

]

= −(1 − A)2J1

= −8A2J−2 + 4A(3A − 1)J−1 + −2√

f1

ξ

∣∣∣∣1

0

.

The only diverging part is the last one. Combination withthe earlier-diverging term gives

limε→0

(−2

√f1

ξ

∣∣∣∣1

ε

− 2(1 − A)√

ε−1 − 1

)= −2

√f1(1) = 0.

Let

It = I1 + I2. (D23)

Now

It = 2(2 + A)π − 4(1 − A)J0 − 12A(1 − 3A)J−1 − 24A2J−2.

The integrals Js are standard elliptic integrals and can betransformed to Legendre normal form with a change of in-tegration variables. We have f1 = g1g2, where g2 = ξ (1 − ξ )and g1(ξ ) > 0 for ξ ∈ [0, 1]. We use the change of variablesξ2 = √

g2/g1. For details, see Ref. [54]. The contribution ofthe quantum potential can be written as q2ρIt/32π , which,after a somewhat tedious calculation, gives

UQP = q2ρ

16

{A + 2 − √

2 − 3A

×[

12C−1A (E (kA) − K (kA)) + (2A + 1)CAK (kA)

π

]},

(D24)

where

K (k) =∫ 1

0dt

[1√

(1 − t2)(1 − k2t2)

](D25)

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HEINONEN, BURNS, AND DUNKEL PHYSICAL REVIEW A 99, 063621 (2019)

0.2 0.4 0.6 0.8 1.0

0.02

0.04

0.06

0.08

0.10

0.000.0

Fit

Solution Eq. (D24)

Relative amplitude 3φ0/ρ

Ave

rage

quan

tum

pot

ential

UQ

P/(q

2ρ)

FIG. 8. Exact solution for the one-mode approximation energy ofthe quantum potential UQP plotted against a polynomial fit showingthat the quartic approximation suffices for analytical calculations.

is the complete elliptic integral of the first kind and

E (k) =∫ 1

0dt

[√1 − k2t2

√1 − t2

](D26)

is the complete elliptic integral of the second kind. Thecomplex coefficients are defined as

kA = i

√√√√ 3A2 +√

(1 − A)3(3A + 1) − 1

−3A2 +√

(1 − A)3(3A + 1) + 1(D27)

and

CA =√

6A2 + 2√

(1 − A)3(3A + 1) − 2

A3. (D28)

The value of UQP at the maximal A = 2/3 can be expressed interms of known constants as

UQP

∣∣A=2/3 =

(1

6−

√3

8π

)q2ρ. (D29)

For more details see the Mathematica notebookelliptic_integral.nb [30].

UQP will be a part of the total internal energy that stillhas to be minimized with respect to q and A. To simplify theminimization procedure we fit a fourth-order polynomial

p4(x) = a2A2 + a3A3 + a4A4 (D30)

against the quantum potential (D24). We find

U = (a2 + a3A + a4A2)A2q2ρ

+ 14 (2(αu + 1) + 3A2(αu + (q2 − 1)2))ρ2, (D31)

where a2 = 0.226 034, a3 = −0.288 956, and a4 =0.401 767. The zeroth coefficients disappears because forzero amplitude the quantum potential is 0. On the otherhand, the first coefficient is 0 because the integral over theperiodic domain of any single oscillating mode gives 0. Thefourth-order fit seems to agree well with the exact solution asshown in Fig. 8.

0.5 1.0 1.5 2.0

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

ρAverage density

Inte

ract

ion p

aram

eter

αu

Exact solution

0.0

Asymptotic

FIG. 9. Uniform-supersolid phase transition line with an asymp-totic expression αu = −0.2234/ρ.

5. Finding the ground state

The ground state of the system is found by minimizingEqs. (D17) and (D31) with respect to the scaler q and theamplitude A = 2φ0/ρ for given system parameters ρ and αu.In both cases, the extrema can be expressed as roots of poly-nomials. The coexistence gap between the uniform superfluidand the supersolid phases is calculated numerically using theanalytical energies via common tangent construction. We findan analytical expression for the phase transition line betweenthe uniform superfluid and the supersolid phases that can beexpressed as an asymptotic expansion for large ρ as αu =−0.2234/ρ. Figure 9 shows the asymptotic expression withthe exact solution. The one-mode analysis gives an upperbound for the SS-NS transition. Minimizing the one-modeenergy given by Eq. (D31) predicts negative densities in theparameter range labeled NS in Fig. 1. In a similar way, the Dphase is characterized by the absence of real solutions to theminimization problem of the one-mode energy. For details onthe ground-state phase diagram see the Mathematica notebookphase_diagram.nb [30].

APPENDIX E: WAVE EQUATION FOR LATTICEVIBRATIONS OF THE SUPERSOLID

We next examine the dynamics of a small displacementfield about the hexagonal ground state described in Sec. D 4 b.To this end, we introduce a deformation of the coordinatesx → x − u(τ, x). The one-mode approximation in Eq. (D18)becomes

ρ(τ, x) = ρ + φ0

3∑j=1

[e−iq j ·u(τ,x)eiq j ·x + c.c.]. (E1)

We write the time evolution equation for the displacementfield u(τ, x) in terms of the effective potential energy U .First, we assume that the relative amplitude φ0/ρ is small andexpand the quantum potential up to fourth order in φ0/ρ. Thisassumption should work near the uniform-supersolid phasetransition line, where the amplitude of the oscillating numberdensity is low. Second, we assume that ∇u is small and expandthe configuration energy U up to second order in u. Third,

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QUANTUM HYDRODYNAMICS FOR SUPERSOLID CRYSTALS … PHYSICAL REVIEW A 99, 063621 (2019)

we assume that the velocity field v is small and keep onlythe linear terms in the dynamical equations. For details ofthe calculation see Ref. [38], where a similar analysis wasperformed.

Now, Eqs.(C2a) and (C2b) give

∂τ u(τ, x) = v(τ, x), ∂τ v(τ, x) = − 1

ρ

δU [u]

δu, (E2)

assuming that φ0 and |q j | = q take the equilibrium value. Thiscan be combined into a single equation,

∂2τ u(τ, x) = − 1

ρ

δU [u]

δu. (E3)

The configuration energy U with the aforementioned approx-imations reduces to

U [u(τ, x)] = 3φ20q2

2

∫dx

×{[

1

4ρ

(1 − φ0

ρ+ 5

φ20

ρ2

)+ (3q2 − 2)

]‖∇u‖2

+ q2((∇ · u)2 + (∇u)T : ∇u) + 2

3‖∇2u‖2

}+U0(αu, φ0, ρ, q). (E4)

Taking the functional derivatives with respect to u and plug-ging in Eq. (E3) leads to

∂2τ u = 3φ2

0q2

ρ

[2q2∇(∇ · u)

+(

3q2 − 2 + 1 − φ0/ρ + 5φ20/ρ

2

4ρ

)∇2u − 2

3∇4u

].

(E5)

Inserting a plane wave exp [i(k · x − ωτ )] gives a dispersionrelation

ω/ω‖ = k/k‖√

1 + (k/k‖)2 (E6)

for longitudinal modes. Here

k2‖ = 3

2

[(5φ2

0/ρ2 − φ0/ρ + 1

)/4ρ − 2 + 5q2

],

ω2‖ = 2q2φ2

0k4‖/ρ. (E7)

At small k the dispersion relation becomes linear and onecan read off the speed of sound:

c2‖ = 3φ2

0q2

ρ

(5q2 − 2 + 1 − φ0/ρ + 5φ2

0/ρ2

4ρ

). (E8)

We can also calculate U0 of Eq. (E4), giving

U0 = 3φ20V�

[q2

4ρ

(1 − φ0/ρ + 5φ2

0/ρ2) + (1 − q2)2

], (E9)

where V� is the volume of the domain. Since φ0/ρ � 1/3,the first part due to the quantum potential penalizes anysystem with q > 0, making the system expand. The secondpart penalizes any variance from q = 1. This part stabilizes thenonuniform pattern. For large values of ρ the pattern formingpart dominates and q ≈ 1.

APPENDIX F: NO PATTERN FORMING FOR LOW-LEVELEXPANSIONS WITH REPULSIVE CONTACT

INTERACTION

Here we show that the low-level expansions of the inter-action term u will only give uniform ground-state solutions.The effective potential energy functional for the second-orderexpansion is

Un[n] =∫

dr[

h2

8m

|∇n|2n

+ g0

2n2 + g2

2|∇n|2

], (F1)

where we use the subscripted Un here to emphasize that theenergy is in SI units. We assume here that g0, g2 � 0. Thefunctional derivative is given by

δUn

δn= g0n − g2∇2n + h2

8m

(−2∇2n

n+ |∇n|2

n2

). (F2)

We want to minimize Un with respect to a conservationconstraint for n. This can be achieved by minimizing

Un[n] = Un[n] − μ

∫dr[n(t, r) − n] (F3)

with respect to n. Here n is a given average particle numberand μ is the chemical potential. Extrema of Un fulfill thecondition

δUn

δn= μ, (F4)

i.e., the functional derivative is a constant. Inserting a uniformn = n in Eq. (F2) gives g0n = μ, showing that n = n is anextremum of Un. We show that this extremum is a globalminimum by showing that the energy Un is globally convex.

First, some prerequisites [55]:(i) A functional is convex iff its second variation is non-

negative for all test functions ϕ.(ii) The sum of convex functionals is convex.The second variation is given by

δ2ϕUn[n] = lim

ε1,ε2→0

d2

dε1dε2Un[n + ε1ϕ + ε2ϕ]. (F5)

The variations for the parts proportional to g0 and g2 can becalculated easily, giving

g0

∫dr[ϕ(t, r)2], g2

∫dr |∇ϕ(t, r)|2,

respectively. Both are clearly nonnegative, implying that thecorresponding parts in U are convex. For the last part we have

|∇n + (ε1 + ε2)∇ϕ|2n + (ε1 + ε2)ϕ

= |∇n + (ε1 + ε2)∇ϕ|2∞∑

k=0

(ε1 + ε2)kϕk

nk+1

= lot + 2ε1ε2[n∇ϕ − ϕ∇n]2

n3+ hot,

where ‘lot’ stands for lower-order terms and ‘hot’ for higher-order terms. This gives the second variation

h2

4m

∫dr{

[n∇ϕ − ϕ∇n]2

n3

}� 0, (F6)

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HEINONEN, BURNS, AND DUNKEL PHYSICAL REVIEW A 99, 063621 (2019)

completing the proof. Setting g2 = 0 proves the same for thezeroth-order expansion.

APPENDIX G: PARAMETERS FORRYDBERG-DRESSED BECS

Here we calculate the interaction parameters q0, ur, and ue

for a model system for Rydberg-dressed BECs described inRef. [27]. The calculation presented here can be repeated forany radially symmetric interatomic potential u whose Fouriertransform’s smallest extremum is a minimum.

We use parameters defined in Fig. 2 in Ref. [27]. Theinteraction potential is

u(r) = usw(r) + uRB(r), (G1)

where usw(r) = gδ(0) is the collision part due to S-wavescattering and

uRB(r) = C6

r6 + R6c

(G2)

is the long-range interaction of two excited alkaline Rydbergatoms. C6, Rc, and g are system parameters.

We calculate the Fourier transform

u(k) =∫ ∞

0

∫ 2π

0dθdr[reikr cos(θ )u(r)]

= g +∫ ∞

0dr

[πrC6rJ0(kr)

R6c + r6

], (G3)

where J0 is the zeroth Bessel function of the first kind. Theremaining integral can be calculated numerically or expressedin terms of the Meijer G function [56] as

u(k) = π

3C6R−4

c G4,00,6

((Rck6

)6; 0, 1

3 , 23 , 2

3 , 0, 13

)+ g. (G4)

In order to obtain the parameters we fit the polynomial

uf(k) = ur + ue(k2 − q2

0

)2(G5)

to the solution u. There are many ways to perform the fitting.This problem has been studied in the context of approximatingdirect pair correlation functions in the classical density func-tional theory of freezing with simpler energy functionals [18].Here we will fit uf by matching q0 to the smallest minimumkm of u and by ensuring that the energy of the uniform andthe supersolid solutions will be correct, i.e., uf(0) = u(0) anduf(q0) = u(q0). We find

q0Rc = q0 ≈ 4.8202,

R4c

C6(ur − g) = 2R4

c

C6u(q0) = ur, (G6)

(q0Rc)4ue

C6= 2R4

c

C6(u(0) − u(q0)) = ue.

The approximate values for these parameters are ur ≈−0.170 23 and ue ≈ 3.9690.

We set g = 0 in accordance with Ref. [27]. The αu param-eter can be calculated as

αu = ur

q40ue

= u(q0)

u(0) − u(q0)= ur

ue, (G7)

TABLE II. Parameter values for Rydberg-dressed BECs.

AlternativeParameter expression Value unit

l√

h/(mωtr ) 0.972674 μmrc Rc/l 2.65c6 C6m3ω2/h4 2.45N – 104

q0 q0/Rc 1.87005 μm−1

ue ueC6/(R4cq4

0 ) 0.197182399 h2/(mq40 )

ur urC6/(R4c ) −0.008456958 h2/m

giving αu ≈ −0.042 889. We also need the average density.In Ref. [27] the authors study a system of 104 Rb atoms in avolume of approximately V ≈ 9R2

c . We have

ρ = mueq20

h2 n = m

R2c h2

ue

q20

N

VC6

= m

R2c h2

ue

q20

N

9R2c

h4c6

m3ω2tr

= h2

m2ω2tr

1

R4c

ue

q20

Nc6

9

=(

l

rc

)4 ue

q20

Nc6

9≈ 9.4. (G8)

Here ωtr/2π = 125 Hz is the trapping frequency of a har-monic potential, and l = √

h/mωtr is the characteristic lengthscale for this trapping potential. The lowercase characters areused for units defined by ωtr and a tilde is used for unitsdefined by C6 and Rc. For parameters, see Table II.

The one-mode approximation predicts the phase transitionat αu ≈ −0.024 539 8, implying that these parameters shouldbe well within the solid regime. The nearest-neighbor distancefor the hexagonal lattice is

ann = 4π

q0

√3, (G9)

which, for the parameters in [27], gives ann ≈ 1.505 16 andRc = 3.879 69 μm.

Figure 10 shows the exact solution for uRB and severaldifferent fits. We use a fitting procedure matching the values ofthe fitted function with the exact solution at k = 0 and k = q0.We also ensure that q0 minimizes uRB. Other fits shown inFig. 10 include matching the curvature and the magnitude atk = km = q0. This would most likely approximate the elasticresponse of the solid phase better since it describes the energywell for modes k close to q0. However, for this system thisfitting method would give an inaccurate estimate for the en-ergy of the uniform state described by u(0). The last fit showsa general least-squares fitting to uRB. This fitting method failsto predict the wavelength of the one-mode approximation, thesolid energy, and the energy of the uniform state, implying thatglobal fitting might not work in this case. For more details seethe Mathematica notebook parameters.nb [30].

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QUANTUM HYDRODYNAMICS FOR SUPERSOLID CRYSTALS … PHYSICAL REVIEW A 99, 063621 (2019)

0.0

Exact solution

1.0

1.0

2.0

3.0

4.0

2.0 3.0 4.0 5.0 6.0-1.0

Match curvature

Least squares fit

Match values

Sca

led F

ourier

tra

nsf

orm

r4 c/α

cu

RB

Scaled wave vector rck

FIG. 10. Rydberg contribution of the Fourier-transformed inter-action potential uRB and different fitting polynomials.

APPENDIX H: NUMERICAL METHODS

1. Ground-state determination

The ground state of the system is found by time-evolvingthe nonconserved dissipative dynamical equation [Eq. (D1)]except in the case of the quantum droplet phase, for whichwe used overdamped dissipative wave dynamics [Eq. (D9)].In general, the performance of the density formalism is betterbecause the high-order derivatives appear linearly in the dy-namical equations. However, the density formalism becomesnumerically unfeasible whenever the one-particle density ρ

is close to 0 due to difficulties in evaluating the quantumpotential Q.

For the ground-state calculations we impose doubly pe-riodic boundary conditions and calculate the derivatives inFourier basis. We use a semiimplicit time-stepping algorithmthat treats the linear parts of the equation implicitly and thenonlinear parts explicitly [57]. To ensure numerical stabilitywe require that the difference of the total energy �E be-tween time steps is nonpositive. This difference is also usedas a stopping condition: whenever �E < 10−12 the iterativesearch for the ground state is stopped.

The simulation domain is a rectangle of size(4πq−1/

√3, 4πq−1). For the periodic ground states the

size of the domain has to be minimized in order to obtainthe physical ground state. This is done as follows: Let somelowest mode of the ground state of a given domain beexp(iq · x). Now q = |q| defines the size of the domain. Dueto the effect of the quantum potential the q that minimizesthe ground-state energy is usually a little bit less than 1. Inorder to find the minimal q we define an iterative process.We make an initial guess for q and calculate a number ofpoints around it. This gives us a chart (qn, En) to which wefit a parabola. We calculate analytically the minimum of thisparabola, define a new set of qn around this minimum, andrepeat the process. Let q(i)

m be the minimal q calculated fromthe fitted parabola at the ith iteration. We stop the iterationwhen |q(i)

m − q(i−1)m | is less than 10−5.

2. Lattice vibration simulations

Lattice vibrations are studied by simulating the full dynam-ics in the hydrodynamic formulation using the spectral PDEsolver DEDALUS [58]. Since the mean-field potential terms arelinear in the density ρ, they can be integrated implicitly in thehydrodynamic formulation, eliminating the stiff time-steppingrestriction that would arise from integrating the higher-orderpotential terms explicitly. Since spectral methods have no nu-merical dissipation, we regularize the equations by insertinga viscositylike term ∝∇2v in the velocity equation, whichsmooths small-scale variations and slowly damps velocityperturbations.

First, the ground state of the periodic domain is calculatedafter which the vibration simulations are initialized with alow-amplitude plane-wave velocity signal. The system wasthen evolved for ∼103 temporal units. The total kinetic energyof the system is observed to undergo oscillations which decaydue to the added viscous term. The square of an exponentiallydecaying sinusoid is fit to the time series of the kinetic energyto determine the oscillation frequency of the velocity signal.The imposed perturbation wavelength and this fit frequencyare used to assess the agreement between the simulations andthe analytical dispersion relation.

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